Algebraicity of <i>L</i>-values attached to quaternionic modular forms

Algebraicity of Singular Values and Arithmetic of Principal Modular Forms

In this thesis, we prove several results on the algebraicity of values of certain modular functions. More precisely, let f(z) be a modular form for the congruence group Γ0(N). We show that f(τ) is an element in the ring class field for the order Oτ corresponding to τ whenever τ is the fixed point of a specific class of matrices in the normalizer of Γ0(N), and provide the explicit Galois action ...

Vector valued hermitian and quaternionic modular forms

Quaternionic Manin symbols, Brandt matrices, and Hilbert modular forms

In this paper, we propose a generalization of the algorithm developed in [4]. Along the way, we also develop a theory of quaternionic M -symbols whose definition bears some resemblance with the classical M -symbols, except for their combinatorial nature. The theory gives a more efficient way to compute Hilbert modular forms over totally fields, especially quadratic fields, and we have illustrat...

Algebraicity of Harmonic Maass Forms

In 1947 D. H. Lehmer conjectured that Ramanujan’s tau-function never vanishes. In the 1980s, B. Gross and D. Zagier proved a deep formula expressing the central derivative of suitable Hasse-Weil L-functions in terms of the Neron-Tate height of a Heegner point. This expository article describes recent work (with J. H. Bruinier and R. Rhoades) which reformulates both topics in terms of the algebr...

Intersections of Higher-weight Cycles over Quaternionic Modular Surfaces and Modular Forms of Nebentypus

In 1976 Hirzebruch and Zagier [6] computed the pairwise intersection multiplicities for a family of algebraic cycles (T£)n(EN in the Hubert modular surface associated to Q(yp), where p is a prime congruent to 1 mod 4, and showed that the generating function for those intersection multiplicities Yln°=oC^m 'T£)e[nr] was an elliptic modular form of weight 2 and Nebentypus for To(p). Shortly afterw...